Chain map

Summary

A chain map is a morphism between two chain complexes: a family of maps $f_n \colon C_n \to D_n$, one in each degree, that commute with the differentials. Chain complexes and chain maps form a category, which is abelian when the underlying category is abelian. Every chain map induces maps on homology, and continuous maps of spaces induce chain maps of singular chains.